Properties

Label 197.10.a.b.1.20
Level $197$
Weight $10$
Character 197.1
Self dual yes
Analytic conductor $101.462$
Analytic rank $0$
Dimension $76$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [197,10,Mod(1,197)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(197, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("197.1");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 197 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 197.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(101.462059724\)
Analytic rank: \(0\)
Dimension: \(76\)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.20
Character \(\chi\) \(=\) 197.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-25.4393 q^{2} +83.7395 q^{3} +135.160 q^{4} -347.412 q^{5} -2130.28 q^{6} +1070.50 q^{7} +9586.56 q^{8} -12670.7 q^{9} +O(q^{10})\) \(q-25.4393 q^{2} +83.7395 q^{3} +135.160 q^{4} -347.412 q^{5} -2130.28 q^{6} +1070.50 q^{7} +9586.56 q^{8} -12670.7 q^{9} +8837.93 q^{10} -65986.5 q^{11} +11318.2 q^{12} +13095.3 q^{13} -27232.8 q^{14} -29092.1 q^{15} -313078. q^{16} -376745. q^{17} +322334. q^{18} +294947. q^{19} -46956.1 q^{20} +89643.0 q^{21} +1.67865e6 q^{22} +1.95882e6 q^{23} +802774. q^{24} -1.83243e6 q^{25} -333135. q^{26} -2.70928e6 q^{27} +144688. q^{28} -6.45398e6 q^{29} +740083. q^{30} -2.48027e6 q^{31} +3.05617e6 q^{32} -5.52568e6 q^{33} +9.58415e6 q^{34} -371904. q^{35} -1.71257e6 q^{36} +9.56186e6 q^{37} -7.50326e6 q^{38} +1.09659e6 q^{39} -3.33049e6 q^{40} +2.55441e7 q^{41} -2.28046e6 q^{42} +2.04501e7 q^{43} -8.91872e6 q^{44} +4.40195e6 q^{45} -4.98310e7 q^{46} +2.39334e7 q^{47} -2.62170e7 q^{48} -3.92076e7 q^{49} +4.66158e7 q^{50} -3.15484e7 q^{51} +1.76996e6 q^{52} -8.96199e7 q^{53} +6.89223e7 q^{54} +2.29245e7 q^{55} +1.02624e7 q^{56} +2.46987e7 q^{57} +1.64185e8 q^{58} -1.28977e8 q^{59} -3.93208e6 q^{60} +1.09979e8 q^{61} +6.30965e7 q^{62} -1.35640e7 q^{63} +8.25489e7 q^{64} -4.54946e6 q^{65} +1.40570e8 q^{66} -2.48183e8 q^{67} -5.09208e7 q^{68} +1.64030e8 q^{69} +9.46099e6 q^{70} -3.36200e8 q^{71} -1.21469e8 q^{72} -7.56846e7 q^{73} -2.43247e8 q^{74} -1.53447e8 q^{75} +3.98650e7 q^{76} -7.06385e7 q^{77} -2.78966e7 q^{78} +4.76941e7 q^{79} +1.08767e8 q^{80} +2.25237e7 q^{81} -6.49825e8 q^{82} +1.87443e8 q^{83} +1.21161e7 q^{84} +1.30886e8 q^{85} -5.20238e8 q^{86} -5.40452e8 q^{87} -6.32584e8 q^{88} +3.24624e8 q^{89} -1.11983e8 q^{90} +1.40185e7 q^{91} +2.64753e8 q^{92} -2.07697e8 q^{93} -6.08849e8 q^{94} -1.02468e8 q^{95} +2.55922e8 q^{96} +3.89600e7 q^{97} +9.97416e8 q^{98} +8.36095e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 76 q + 48 q^{2} + 890 q^{3} + 20736 q^{4} + 5171 q^{5} + 2688 q^{6} + 38986 q^{7} + 36507 q^{8} + 518318 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 76 q + 48 q^{2} + 890 q^{3} + 20736 q^{4} + 5171 q^{5} + 2688 q^{6} + 38986 q^{7} + 36507 q^{8} + 518318 q^{9} + 121093 q^{10} + 120464 q^{11} + 415744 q^{12} + 480131 q^{13} + 330849 q^{14} + 544874 q^{15} + 5963776 q^{16} + 942582 q^{17} + 1483945 q^{18} + 3097319 q^{19} + 3237739 q^{20} + 889076 q^{21} + 3791921 q^{22} + 5139200 q^{23} + 1999533 q^{24} + 34080519 q^{25} + 2791454 q^{26} + 24386486 q^{27} + 20891166 q^{28} + 6886818 q^{29} + 14171083 q^{30} + 28002851 q^{31} + 16857332 q^{32} + 30921422 q^{33} + 33506194 q^{34} + 16271736 q^{35} + 151430458 q^{36} + 55950976 q^{37} + 62370882 q^{38} - 11569592 q^{39} + 129854766 q^{40} + 14990859 q^{41} + 82216531 q^{42} + 169867467 q^{43} + 41872434 q^{44} + 205007649 q^{45} + 144032301 q^{46} + 78743342 q^{47} + 156250562 q^{48} + 533861890 q^{49} + 626841163 q^{50} + 477099244 q^{51} + 560784114 q^{52} + 188670216 q^{53} + 525901687 q^{54} + 298497914 q^{55} + 56575048 q^{56} + 213972590 q^{57} + 338315251 q^{58} + 208222151 q^{59} - 615921507 q^{60} - 233556134 q^{61} - 399368105 q^{62} + 329825056 q^{63} + 876517017 q^{64} - 840557006 q^{65} - 2482481592 q^{66} + 1210808414 q^{67} - 1266757099 q^{68} + 327801786 q^{69} - 546384313 q^{70} + 345300221 q^{71} - 1549481681 q^{72} + 1192286460 q^{73} - 1471133595 q^{74} + 761630676 q^{75} - 398699826 q^{76} - 101106252 q^{77} - 2609825943 q^{78} + 955627631 q^{79} + 1059617770 q^{80} + 3387041436 q^{81} + 1062705523 q^{82} + 1538917201 q^{83} + 1394513218 q^{84} + 225481100 q^{85} + 701644810 q^{86} + 1758812842 q^{87} + 3151474875 q^{88} + 855413630 q^{89} + 6070671455 q^{90} + 4652436248 q^{91} + 8082863606 q^{92} + 3462095982 q^{93} + 2660342117 q^{94} + 1036805508 q^{95} + 12370989029 q^{96} + 6393874545 q^{97} + 7510976010 q^{98} + 8731109606 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −25.4393 −1.12427 −0.562135 0.827045i \(-0.690020\pi\)
−0.562135 + 0.827045i \(0.690020\pi\)
\(3\) 83.7395 0.596877 0.298438 0.954429i \(-0.403534\pi\)
0.298438 + 0.954429i \(0.403534\pi\)
\(4\) 135.160 0.263984
\(5\) −347.412 −0.248588 −0.124294 0.992245i \(-0.539667\pi\)
−0.124294 + 0.992245i \(0.539667\pi\)
\(6\) −2130.28 −0.671051
\(7\) 1070.50 0.168517 0.0842587 0.996444i \(-0.473148\pi\)
0.0842587 + 0.996444i \(0.473148\pi\)
\(8\) 9586.56 0.827481
\(9\) −12670.7 −0.643738
\(10\) 8837.93 0.279480
\(11\) −65986.5 −1.35890 −0.679451 0.733721i \(-0.737782\pi\)
−0.679451 + 0.733721i \(0.737782\pi\)
\(12\) 11318.2 0.157566
\(13\) 13095.3 0.127166 0.0635828 0.997977i \(-0.479747\pi\)
0.0635828 + 0.997977i \(0.479747\pi\)
\(14\) −27232.8 −0.189459
\(15\) −29092.1 −0.148376
\(16\) −313078. −1.19430
\(17\) −376745. −1.09403 −0.547013 0.837124i \(-0.684235\pi\)
−0.547013 + 0.837124i \(0.684235\pi\)
\(18\) 322334. 0.723736
\(19\) 294947. 0.519222 0.259611 0.965713i \(-0.416406\pi\)
0.259611 + 0.965713i \(0.416406\pi\)
\(20\) −46956.1 −0.0656232
\(21\) 89643.0 0.100584
\(22\) 1.67865e6 1.52777
\(23\) 1.95882e6 1.45955 0.729774 0.683688i \(-0.239625\pi\)
0.729774 + 0.683688i \(0.239625\pi\)
\(24\) 802774. 0.493904
\(25\) −1.83243e6 −0.938204
\(26\) −333135. −0.142969
\(27\) −2.70928e6 −0.981109
\(28\) 144688. 0.0444859
\(29\) −6.45398e6 −1.69448 −0.847240 0.531211i \(-0.821737\pi\)
−0.847240 + 0.531211i \(0.821737\pi\)
\(30\) 740083. 0.166815
\(31\) −2.48027e6 −0.482360 −0.241180 0.970480i \(-0.577534\pi\)
−0.241180 + 0.970480i \(0.577534\pi\)
\(32\) 3.05617e6 0.515231
\(33\) −5.52568e6 −0.811097
\(34\) 9.58415e6 1.22998
\(35\) −371904. −0.0418914
\(36\) −1.71257e6 −0.169937
\(37\) 9.56186e6 0.838754 0.419377 0.907812i \(-0.362249\pi\)
0.419377 + 0.907812i \(0.362249\pi\)
\(38\) −7.50326e6 −0.583745
\(39\) 1.09659e6 0.0759022
\(40\) −3.33049e6 −0.205702
\(41\) 2.55441e7 1.41177 0.705884 0.708327i \(-0.250550\pi\)
0.705884 + 0.708327i \(0.250550\pi\)
\(42\) −2.28046e6 −0.113084
\(43\) 2.04501e7 0.912196 0.456098 0.889929i \(-0.349247\pi\)
0.456098 + 0.889929i \(0.349247\pi\)
\(44\) −8.91872e6 −0.358728
\(45\) 4.40195e6 0.160025
\(46\) −4.98310e7 −1.64093
\(47\) 2.39334e7 0.715424 0.357712 0.933832i \(-0.383557\pi\)
0.357712 + 0.933832i \(0.383557\pi\)
\(48\) −2.62170e7 −0.712848
\(49\) −3.92076e7 −0.971602
\(50\) 4.66158e7 1.05480
\(51\) −3.15484e7 −0.652998
\(52\) 1.76996e6 0.0335697
\(53\) −8.96199e7 −1.56014 −0.780069 0.625694i \(-0.784816\pi\)
−0.780069 + 0.625694i \(0.784816\pi\)
\(54\) 6.89223e7 1.10303
\(55\) 2.29245e7 0.337806
\(56\) 1.02624e7 0.139445
\(57\) 2.46987e7 0.309911
\(58\) 1.64185e8 1.90505
\(59\) −1.28977e8 −1.38572 −0.692862 0.721070i \(-0.743650\pi\)
−0.692862 + 0.721070i \(0.743650\pi\)
\(60\) −3.93208e6 −0.0391689
\(61\) 1.09979e8 1.01701 0.508504 0.861059i \(-0.330199\pi\)
0.508504 + 0.861059i \(0.330199\pi\)
\(62\) 6.30965e7 0.542304
\(63\) −1.35640e7 −0.108481
\(64\) 8.25489e7 0.615037
\(65\) −4.54946e6 −0.0316118
\(66\) 1.40570e8 0.911892
\(67\) −2.48183e8 −1.50465 −0.752324 0.658793i \(-0.771067\pi\)
−0.752324 + 0.658793i \(0.771067\pi\)
\(68\) −5.09208e7 −0.288805
\(69\) 1.64030e8 0.871170
\(70\) 9.46099e6 0.0470972
\(71\) −3.36200e8 −1.57013 −0.785064 0.619414i \(-0.787370\pi\)
−0.785064 + 0.619414i \(0.787370\pi\)
\(72\) −1.21469e8 −0.532681
\(73\) −7.56846e7 −0.311928 −0.155964 0.987763i \(-0.549848\pi\)
−0.155964 + 0.987763i \(0.549848\pi\)
\(74\) −2.43247e8 −0.942986
\(75\) −1.53447e8 −0.559992
\(76\) 3.98650e7 0.137066
\(77\) −7.06385e7 −0.228999
\(78\) −2.78966e7 −0.0853346
\(79\) 4.76941e7 0.137766 0.0688831 0.997625i \(-0.478056\pi\)
0.0688831 + 0.997625i \(0.478056\pi\)
\(80\) 1.08767e8 0.296887
\(81\) 2.25237e7 0.0581375
\(82\) −6.49825e8 −1.58721
\(83\) 1.87443e8 0.433528 0.216764 0.976224i \(-0.430450\pi\)
0.216764 + 0.976224i \(0.430450\pi\)
\(84\) 1.21161e7 0.0265526
\(85\) 1.30886e8 0.271961
\(86\) −5.20238e8 −1.02556
\(87\) −5.40452e8 −1.01140
\(88\) −6.32584e8 −1.12447
\(89\) 3.24624e8 0.548436 0.274218 0.961668i \(-0.411581\pi\)
0.274218 + 0.961668i \(0.411581\pi\)
\(90\) −1.11983e8 −0.179912
\(91\) 1.40185e7 0.0214296
\(92\) 2.64753e8 0.385297
\(93\) −2.07697e8 −0.287910
\(94\) −6.08849e8 −0.804330
\(95\) −1.02468e8 −0.129072
\(96\) 2.55922e8 0.307529
\(97\) 3.89600e7 0.0446834 0.0223417 0.999750i \(-0.492888\pi\)
0.0223417 + 0.999750i \(0.492888\pi\)
\(98\) 9.97416e8 1.09234
\(99\) 8.36095e8 0.874777
\(100\) −2.47671e8 −0.247671
\(101\) −1.12674e9 −1.07740 −0.538700 0.842498i \(-0.681084\pi\)
−0.538700 + 0.842498i \(0.681084\pi\)
\(102\) 8.02571e8 0.734147
\(103\) −8.18572e8 −0.716621 −0.358310 0.933603i \(-0.616647\pi\)
−0.358310 + 0.933603i \(0.616647\pi\)
\(104\) 1.25539e8 0.105227
\(105\) −3.11430e7 −0.0250040
\(106\) 2.27987e9 1.75402
\(107\) 1.13627e8 0.0838023 0.0419012 0.999122i \(-0.486659\pi\)
0.0419012 + 0.999122i \(0.486659\pi\)
\(108\) −3.66186e8 −0.258997
\(109\) 2.43020e9 1.64901 0.824506 0.565854i \(-0.191453\pi\)
0.824506 + 0.565854i \(0.191453\pi\)
\(110\) −5.83184e8 −0.379786
\(111\) 8.00705e8 0.500632
\(112\) −3.35149e8 −0.201260
\(113\) 8.33615e8 0.480964 0.240482 0.970654i \(-0.422694\pi\)
0.240482 + 0.970654i \(0.422694\pi\)
\(114\) −6.28319e8 −0.348424
\(115\) −6.80516e8 −0.362826
\(116\) −8.72318e8 −0.447315
\(117\) −1.65926e8 −0.0818614
\(118\) 3.28108e9 1.55793
\(119\) −4.03305e8 −0.184362
\(120\) −2.78893e8 −0.122779
\(121\) 1.99627e9 0.846614
\(122\) −2.79779e9 −1.14339
\(123\) 2.13905e9 0.842651
\(124\) −3.35233e8 −0.127335
\(125\) 1.31515e9 0.481814
\(126\) 3.45058e8 0.121962
\(127\) 1.98972e9 0.678695 0.339348 0.940661i \(-0.389794\pi\)
0.339348 + 0.940661i \(0.389794\pi\)
\(128\) −3.66475e9 −1.20670
\(129\) 1.71248e9 0.544469
\(130\) 1.15735e8 0.0355402
\(131\) 1.15149e9 0.341618 0.170809 0.985304i \(-0.445362\pi\)
0.170809 + 0.985304i \(0.445362\pi\)
\(132\) −7.46849e8 −0.214117
\(133\) 3.15740e8 0.0874979
\(134\) 6.31360e9 1.69163
\(135\) 9.41237e8 0.243892
\(136\) −3.61169e9 −0.905285
\(137\) 3.87948e9 0.940873 0.470437 0.882434i \(-0.344096\pi\)
0.470437 + 0.882434i \(0.344096\pi\)
\(138\) −4.17282e9 −0.979431
\(139\) 1.00219e9 0.227712 0.113856 0.993497i \(-0.463680\pi\)
0.113856 + 0.993497i \(0.463680\pi\)
\(140\) −5.02665e7 −0.0110587
\(141\) 2.00417e9 0.427020
\(142\) 8.55271e9 1.76525
\(143\) −8.64112e8 −0.172806
\(144\) 3.96691e9 0.768814
\(145\) 2.24219e9 0.421227
\(146\) 1.92537e9 0.350692
\(147\) −3.28323e9 −0.579926
\(148\) 1.29238e9 0.221418
\(149\) 2.90922e9 0.483546 0.241773 0.970333i \(-0.422271\pi\)
0.241773 + 0.970333i \(0.422271\pi\)
\(150\) 3.90358e9 0.629582
\(151\) 8.62779e9 1.35053 0.675264 0.737576i \(-0.264030\pi\)
0.675264 + 0.737576i \(0.264030\pi\)
\(152\) 2.82753e9 0.429646
\(153\) 4.77363e9 0.704266
\(154\) 1.79700e9 0.257456
\(155\) 8.61676e8 0.119909
\(156\) 1.48215e8 0.0200370
\(157\) 1.03295e10 1.35685 0.678426 0.734669i \(-0.262662\pi\)
0.678426 + 0.734669i \(0.262662\pi\)
\(158\) −1.21331e9 −0.154887
\(159\) −7.50472e9 −0.931210
\(160\) −1.06175e9 −0.128080
\(161\) 2.09691e9 0.245959
\(162\) −5.72987e8 −0.0653623
\(163\) −8.87803e9 −0.985083 −0.492541 0.870289i \(-0.663932\pi\)
−0.492541 + 0.870289i \(0.663932\pi\)
\(164\) 3.45254e9 0.372684
\(165\) 1.91969e9 0.201629
\(166\) −4.76842e9 −0.487402
\(167\) 1.35813e10 1.35120 0.675598 0.737270i \(-0.263885\pi\)
0.675598 + 0.737270i \(0.263885\pi\)
\(168\) 8.59368e8 0.0832314
\(169\) −1.04330e10 −0.983829
\(170\) −3.32965e9 −0.305758
\(171\) −3.73719e9 −0.334243
\(172\) 2.76404e9 0.240805
\(173\) 1.38587e10 1.17629 0.588145 0.808755i \(-0.299858\pi\)
0.588145 + 0.808755i \(0.299858\pi\)
\(174\) 1.37488e10 1.13708
\(175\) −1.96161e9 −0.158104
\(176\) 2.06589e10 1.62293
\(177\) −1.08004e10 −0.827106
\(178\) −8.25823e9 −0.616591
\(179\) 6.49482e9 0.472855 0.236428 0.971649i \(-0.424023\pi\)
0.236428 + 0.971649i \(0.424023\pi\)
\(180\) 5.94967e8 0.0422442
\(181\) 9.54862e9 0.661282 0.330641 0.943757i \(-0.392735\pi\)
0.330641 + 0.943757i \(0.392735\pi\)
\(182\) −3.56621e8 −0.0240927
\(183\) 9.20956e9 0.607029
\(184\) 1.87783e10 1.20775
\(185\) −3.32190e9 −0.208504
\(186\) 5.28366e9 0.323688
\(187\) 2.48601e10 1.48667
\(188\) 3.23483e9 0.188861
\(189\) −2.90028e9 −0.165334
\(190\) 2.60672e9 0.145112
\(191\) 1.84838e10 1.00494 0.502470 0.864594i \(-0.332425\pi\)
0.502470 + 0.864594i \(0.332425\pi\)
\(192\) 6.91260e9 0.367101
\(193\) −2.87866e10 −1.49342 −0.746711 0.665149i \(-0.768368\pi\)
−0.746711 + 0.665149i \(0.768368\pi\)
\(194\) −9.91116e8 −0.0502362
\(195\) −3.80969e8 −0.0188684
\(196\) −5.29930e9 −0.256487
\(197\) 1.50614e9 0.0712470
\(198\) −2.12697e10 −0.983486
\(199\) 2.01842e10 0.912374 0.456187 0.889884i \(-0.349215\pi\)
0.456187 + 0.889884i \(0.349215\pi\)
\(200\) −1.75667e10 −0.776346
\(201\) −2.07827e10 −0.898089
\(202\) 2.86635e10 1.21129
\(203\) −6.90897e9 −0.285549
\(204\) −4.26408e9 −0.172381
\(205\) −8.87433e9 −0.350948
\(206\) 2.08239e10 0.805675
\(207\) −2.48196e10 −0.939567
\(208\) −4.09984e9 −0.151873
\(209\) −1.94625e10 −0.705571
\(210\) 7.92258e8 0.0281112
\(211\) 5.18101e10 1.79947 0.899733 0.436441i \(-0.143761\pi\)
0.899733 + 0.436441i \(0.143761\pi\)
\(212\) −1.21130e10 −0.411851
\(213\) −2.81532e10 −0.937173
\(214\) −2.89061e9 −0.0942165
\(215\) −7.10462e9 −0.226761
\(216\) −2.59727e10 −0.811849
\(217\) −2.65513e9 −0.0812861
\(218\) −6.18228e10 −1.85393
\(219\) −6.33779e9 −0.186183
\(220\) 3.09847e9 0.0891755
\(221\) −4.93358e9 −0.139122
\(222\) −2.03694e10 −0.562846
\(223\) −5.04084e9 −0.136499 −0.0682497 0.997668i \(-0.521741\pi\)
−0.0682497 + 0.997668i \(0.521741\pi\)
\(224\) 3.27162e9 0.0868254
\(225\) 2.32182e10 0.603958
\(226\) −2.12066e10 −0.540734
\(227\) −7.67396e9 −0.191824 −0.0959121 0.995390i \(-0.530577\pi\)
−0.0959121 + 0.995390i \(0.530577\pi\)
\(228\) 3.33827e9 0.0818116
\(229\) 3.85554e10 0.926457 0.463229 0.886239i \(-0.346691\pi\)
0.463229 + 0.886239i \(0.346691\pi\)
\(230\) 1.73119e10 0.407914
\(231\) −5.91523e9 −0.136684
\(232\) −6.18715e10 −1.40215
\(233\) 3.08810e10 0.686419 0.343210 0.939259i \(-0.388486\pi\)
0.343210 + 0.939259i \(0.388486\pi\)
\(234\) 4.22106e9 0.0920343
\(235\) −8.31474e9 −0.177846
\(236\) −1.74324e10 −0.365809
\(237\) 3.99388e9 0.0822294
\(238\) 1.02598e10 0.207273
\(239\) 6.43440e10 1.27561 0.637804 0.770199i \(-0.279843\pi\)
0.637804 + 0.770199i \(0.279843\pi\)
\(240\) 9.10808e9 0.177205
\(241\) −5.98359e10 −1.14258 −0.571289 0.820749i \(-0.693556\pi\)
−0.571289 + 0.820749i \(0.693556\pi\)
\(242\) −5.07838e10 −0.951823
\(243\) 5.52129e10 1.01581
\(244\) 1.48647e10 0.268474
\(245\) 1.36212e10 0.241528
\(246\) −5.44160e10 −0.947368
\(247\) 3.86241e9 0.0660271
\(248\) −2.37773e10 −0.399144
\(249\) 1.56963e10 0.258763
\(250\) −3.34565e10 −0.541689
\(251\) −5.21882e10 −0.829928 −0.414964 0.909838i \(-0.636206\pi\)
−0.414964 + 0.909838i \(0.636206\pi\)
\(252\) −1.83330e9 −0.0286373
\(253\) −1.29255e11 −1.98338
\(254\) −5.06171e10 −0.763037
\(255\) 1.09603e10 0.162327
\(256\) 5.09637e10 0.741619
\(257\) 3.63401e9 0.0519622 0.0259811 0.999662i \(-0.491729\pi\)
0.0259811 + 0.999662i \(0.491729\pi\)
\(258\) −4.35645e10 −0.612130
\(259\) 1.02360e10 0.141345
\(260\) −6.14904e8 −0.00834501
\(261\) 8.17764e10 1.09080
\(262\) −2.92932e10 −0.384071
\(263\) −1.05265e11 −1.35669 −0.678347 0.734742i \(-0.737303\pi\)
−0.678347 + 0.734742i \(0.737303\pi\)
\(264\) −5.29722e10 −0.671167
\(265\) 3.11350e10 0.387831
\(266\) −8.03222e9 −0.0983713
\(267\) 2.71839e10 0.327349
\(268\) −3.35443e10 −0.397203
\(269\) 1.45429e11 1.69343 0.846713 0.532050i \(-0.178578\pi\)
0.846713 + 0.532050i \(0.178578\pi\)
\(270\) −2.39444e10 −0.274200
\(271\) −8.79643e10 −0.990706 −0.495353 0.868692i \(-0.664961\pi\)
−0.495353 + 0.868692i \(0.664961\pi\)
\(272\) 1.17950e11 1.30659
\(273\) 1.17390e9 0.0127908
\(274\) −9.86915e10 −1.05780
\(275\) 1.20916e11 1.27493
\(276\) 2.21703e10 0.229975
\(277\) 1.24181e11 1.26734 0.633672 0.773602i \(-0.281547\pi\)
0.633672 + 0.773602i \(0.281547\pi\)
\(278\) −2.54952e10 −0.256010
\(279\) 3.14268e10 0.310514
\(280\) −3.56528e9 −0.0346643
\(281\) −3.62284e10 −0.346633 −0.173317 0.984866i \(-0.555448\pi\)
−0.173317 + 0.984866i \(0.555448\pi\)
\(282\) −5.09847e10 −0.480086
\(283\) −4.57153e10 −0.423665 −0.211833 0.977306i \(-0.567943\pi\)
−0.211833 + 0.977306i \(0.567943\pi\)
\(284\) −4.54407e10 −0.414489
\(285\) −8.58062e9 −0.0770401
\(286\) 2.19824e10 0.194280
\(287\) 2.73449e10 0.237908
\(288\) −3.87238e10 −0.331674
\(289\) 2.33490e10 0.196892
\(290\) −5.70398e10 −0.473573
\(291\) 3.26249e9 0.0266704
\(292\) −1.02295e10 −0.0823441
\(293\) −8.36775e10 −0.663292 −0.331646 0.943404i \(-0.607604\pi\)
−0.331646 + 0.943404i \(0.607604\pi\)
\(294\) 8.35231e10 0.651994
\(295\) 4.48080e10 0.344474
\(296\) 9.16653e10 0.694053
\(297\) 1.78776e11 1.33323
\(298\) −7.40086e10 −0.543637
\(299\) 2.56513e10 0.185604
\(300\) −2.07398e10 −0.147829
\(301\) 2.18918e10 0.153721
\(302\) −2.19485e11 −1.51836
\(303\) −9.43524e10 −0.643074
\(304\) −9.23413e10 −0.620105
\(305\) −3.82079e10 −0.252816
\(306\) −1.21438e11 −0.791786
\(307\) −1.54276e11 −0.991235 −0.495618 0.868541i \(-0.665058\pi\)
−0.495618 + 0.868541i \(0.665058\pi\)
\(308\) −9.54748e9 −0.0604520
\(309\) −6.85468e10 −0.427734
\(310\) −2.19205e10 −0.134810
\(311\) −1.33921e11 −0.811758 −0.405879 0.913927i \(-0.633034\pi\)
−0.405879 + 0.913927i \(0.633034\pi\)
\(312\) 1.05125e10 0.0628076
\(313\) 2.35267e11 1.38551 0.692757 0.721171i \(-0.256396\pi\)
0.692757 + 0.721171i \(0.256396\pi\)
\(314\) −2.62777e11 −1.52547
\(315\) 4.71228e9 0.0269671
\(316\) 6.44633e9 0.0363681
\(317\) −2.55517e11 −1.42119 −0.710597 0.703599i \(-0.751575\pi\)
−0.710597 + 0.703599i \(0.751575\pi\)
\(318\) 1.90915e11 1.04693
\(319\) 4.25875e11 2.30263
\(320\) −2.86785e10 −0.152891
\(321\) 9.51510e9 0.0500197
\(322\) −5.33440e10 −0.276525
\(323\) −1.11120e11 −0.568042
\(324\) 3.04429e9 0.0153474
\(325\) −2.39962e10 −0.119307
\(326\) 2.25851e11 1.10750
\(327\) 2.03504e11 0.984256
\(328\) 2.44880e11 1.16821
\(329\) 2.56206e10 0.120561
\(330\) −4.88355e10 −0.226685
\(331\) −5.77463e10 −0.264423 −0.132211 0.991222i \(-0.542208\pi\)
−0.132211 + 0.991222i \(0.542208\pi\)
\(332\) 2.53347e10 0.114444
\(333\) −1.21155e11 −0.539938
\(334\) −3.45500e11 −1.51911
\(335\) 8.62216e10 0.374037
\(336\) −2.80652e10 −0.120127
\(337\) 1.88165e11 0.794703 0.397351 0.917667i \(-0.369929\pi\)
0.397351 + 0.917667i \(0.369929\pi\)
\(338\) 2.65409e11 1.10609
\(339\) 6.98065e10 0.287076
\(340\) 1.76905e10 0.0717935
\(341\) 1.63664e11 0.655480
\(342\) 9.50715e10 0.375779
\(343\) −8.51702e10 −0.332249
\(344\) 1.96047e11 0.754825
\(345\) −5.69861e10 −0.216562
\(346\) −3.52556e11 −1.32247
\(347\) −8.77882e10 −0.325052 −0.162526 0.986704i \(-0.551964\pi\)
−0.162526 + 0.986704i \(0.551964\pi\)
\(348\) −7.30474e10 −0.266992
\(349\) 4.79459e11 1.72996 0.864982 0.501803i \(-0.167330\pi\)
0.864982 + 0.501803i \(0.167330\pi\)
\(350\) 4.99021e10 0.177751
\(351\) −3.54788e10 −0.124763
\(352\) −2.01666e11 −0.700149
\(353\) −5.95046e10 −0.203969 −0.101985 0.994786i \(-0.532519\pi\)
−0.101985 + 0.994786i \(0.532519\pi\)
\(354\) 2.74756e11 0.929891
\(355\) 1.16800e11 0.390315
\(356\) 4.38762e10 0.144778
\(357\) −3.37725e10 −0.110042
\(358\) −1.65224e11 −0.531617
\(359\) 6.61235e10 0.210102 0.105051 0.994467i \(-0.466499\pi\)
0.105051 + 0.994467i \(0.466499\pi\)
\(360\) 4.21996e10 0.132418
\(361\) −2.35694e11 −0.730409
\(362\) −2.42910e11 −0.743460
\(363\) 1.67167e11 0.505324
\(364\) 1.89473e9 0.00565708
\(365\) 2.62937e10 0.0775416
\(366\) −2.34285e11 −0.682464
\(367\) 4.31936e11 1.24286 0.621430 0.783470i \(-0.286552\pi\)
0.621430 + 0.783470i \(0.286552\pi\)
\(368\) −6.13262e11 −1.74313
\(369\) −3.23662e11 −0.908809
\(370\) 8.45070e10 0.234415
\(371\) −9.59380e10 −0.262910
\(372\) −2.80722e10 −0.0760035
\(373\) 2.34458e11 0.627155 0.313578 0.949563i \(-0.398472\pi\)
0.313578 + 0.949563i \(0.398472\pi\)
\(374\) −6.32424e11 −1.67142
\(375\) 1.10130e11 0.287583
\(376\) 2.29439e11 0.592000
\(377\) −8.45166e10 −0.215480
\(378\) 7.37812e10 0.185880
\(379\) 1.71270e11 0.426388 0.213194 0.977010i \(-0.431613\pi\)
0.213194 + 0.977010i \(0.431613\pi\)
\(380\) −1.38496e10 −0.0340730
\(381\) 1.66618e11 0.405097
\(382\) −4.70215e11 −1.12982
\(383\) −5.78488e11 −1.37373 −0.686863 0.726787i \(-0.741013\pi\)
−0.686863 + 0.726787i \(0.741013\pi\)
\(384\) −3.06884e11 −0.720251
\(385\) 2.45406e10 0.0569263
\(386\) 7.32312e11 1.67901
\(387\) −2.59118e11 −0.587216
\(388\) 5.26582e9 0.0117957
\(389\) 1.87685e11 0.415581 0.207791 0.978173i \(-0.433373\pi\)
0.207791 + 0.978173i \(0.433373\pi\)
\(390\) 9.69160e9 0.0212131
\(391\) −7.37975e11 −1.59678
\(392\) −3.75867e11 −0.803982
\(393\) 9.64254e10 0.203904
\(394\) −3.83152e10 −0.0801010
\(395\) −1.65695e10 −0.0342470
\(396\) 1.13006e11 0.230927
\(397\) −6.80816e10 −0.137554 −0.0687769 0.997632i \(-0.521910\pi\)
−0.0687769 + 0.997632i \(0.521910\pi\)
\(398\) −5.13473e11 −1.02575
\(399\) 2.64399e10 0.0522254
\(400\) 5.73693e11 1.12049
\(401\) −3.21067e11 −0.620077 −0.310039 0.950724i \(-0.600342\pi\)
−0.310039 + 0.950724i \(0.600342\pi\)
\(402\) 5.28698e11 1.00969
\(403\) −3.24799e10 −0.0613397
\(404\) −1.52290e11 −0.284416
\(405\) −7.82499e9 −0.0144523
\(406\) 1.75760e11 0.321035
\(407\) −6.30954e11 −1.13978
\(408\) −3.02441e11 −0.540344
\(409\) 1.03455e12 1.82808 0.914042 0.405619i \(-0.132944\pi\)
0.914042 + 0.405619i \(0.132944\pi\)
\(410\) 2.25757e11 0.394561
\(411\) 3.24866e11 0.561585
\(412\) −1.10638e11 −0.189176
\(413\) −1.38069e11 −0.233519
\(414\) 6.31394e11 1.05633
\(415\) −6.51198e10 −0.107770
\(416\) 4.00214e10 0.0655197
\(417\) 8.39232e10 0.135916
\(418\) 4.95114e11 0.793253
\(419\) −8.26593e11 −1.31017 −0.655087 0.755554i \(-0.727368\pi\)
−0.655087 + 0.755554i \(0.727368\pi\)
\(420\) −4.20929e9 −0.00660065
\(421\) 7.04223e11 1.09255 0.546274 0.837606i \(-0.316046\pi\)
0.546274 + 0.837606i \(0.316046\pi\)
\(422\) −1.31801e12 −2.02309
\(423\) −3.03253e11 −0.460546
\(424\) −8.59147e11 −1.29098
\(425\) 6.90359e11 1.02642
\(426\) 7.16199e11 1.05364
\(427\) 1.17732e11 0.171384
\(428\) 1.53579e10 0.0221225
\(429\) −7.23603e10 −0.103144
\(430\) 1.80737e11 0.254940
\(431\) 2.75449e11 0.384498 0.192249 0.981346i \(-0.438422\pi\)
0.192249 + 0.981346i \(0.438422\pi\)
\(432\) 8.48216e11 1.17173
\(433\) 1.00904e9 0.00137947 0.000689734 1.00000i \(-0.499780\pi\)
0.000689734 1.00000i \(0.499780\pi\)
\(434\) 6.75447e10 0.0913876
\(435\) 1.87760e11 0.251420
\(436\) 3.28466e11 0.435313
\(437\) 5.77747e11 0.757829
\(438\) 1.61229e11 0.209320
\(439\) 8.24219e11 1.05914 0.529569 0.848267i \(-0.322354\pi\)
0.529569 + 0.848267i \(0.322354\pi\)
\(440\) 2.19767e11 0.279528
\(441\) 4.96788e11 0.625457
\(442\) 1.25507e11 0.156411
\(443\) −1.44616e12 −1.78402 −0.892010 0.452015i \(-0.850705\pi\)
−0.892010 + 0.452015i \(0.850705\pi\)
\(444\) 1.08223e11 0.132159
\(445\) −1.12778e11 −0.136335
\(446\) 1.28236e11 0.153462
\(447\) 2.43616e11 0.288617
\(448\) 8.83685e10 0.103645
\(449\) −2.16663e11 −0.251580 −0.125790 0.992057i \(-0.540146\pi\)
−0.125790 + 0.992057i \(0.540146\pi\)
\(450\) −5.90655e11 −0.679012
\(451\) −1.68557e12 −1.91845
\(452\) 1.12671e11 0.126967
\(453\) 7.22487e11 0.806098
\(454\) 1.95220e11 0.215662
\(455\) −4.87019e9 −0.00532714
\(456\) 2.36776e11 0.256446
\(457\) −7.21024e11 −0.773262 −0.386631 0.922234i \(-0.626361\pi\)
−0.386631 + 0.922234i \(0.626361\pi\)
\(458\) −9.80823e11 −1.04159
\(459\) 1.02071e12 1.07336
\(460\) −9.19785e10 −0.0957802
\(461\) −1.95490e11 −0.201591 −0.100796 0.994907i \(-0.532139\pi\)
−0.100796 + 0.994907i \(0.532139\pi\)
\(462\) 1.50479e11 0.153670
\(463\) 1.48300e12 1.49978 0.749888 0.661564i \(-0.230107\pi\)
0.749888 + 0.661564i \(0.230107\pi\)
\(464\) 2.02060e12 2.02371
\(465\) 7.21563e10 0.0715708
\(466\) −7.85592e11 −0.771721
\(467\) −1.15178e12 −1.12058 −0.560292 0.828295i \(-0.689311\pi\)
−0.560292 + 0.828295i \(0.689311\pi\)
\(468\) −2.24266e10 −0.0216101
\(469\) −2.65679e11 −0.253559
\(470\) 2.11521e11 0.199947
\(471\) 8.64990e11 0.809873
\(472\) −1.23644e12 −1.14666
\(473\) −1.34943e12 −1.23959
\(474\) −1.01602e11 −0.0924481
\(475\) −5.40470e11 −0.487136
\(476\) −5.45106e10 −0.0486687
\(477\) 1.13555e12 1.00432
\(478\) −1.63687e12 −1.43413
\(479\) 1.26546e12 1.09835 0.549174 0.835708i \(-0.314942\pi\)
0.549174 + 0.835708i \(0.314942\pi\)
\(480\) −8.89103e10 −0.0764480
\(481\) 1.25215e11 0.106661
\(482\) 1.52219e12 1.28457
\(483\) 1.75594e11 0.146807
\(484\) 2.69816e11 0.223493
\(485\) −1.35352e10 −0.0111077
\(486\) −1.40458e12 −1.14204
\(487\) 1.09759e11 0.0884217 0.0442108 0.999022i \(-0.485923\pi\)
0.0442108 + 0.999022i \(0.485923\pi\)
\(488\) 1.05432e12 0.841555
\(489\) −7.43442e11 −0.587973
\(490\) −3.46514e11 −0.271543
\(491\) 2.19958e12 1.70794 0.853972 0.520319i \(-0.174187\pi\)
0.853972 + 0.520319i \(0.174187\pi\)
\(492\) 2.89114e11 0.222446
\(493\) 2.43150e12 1.85380
\(494\) −9.82572e10 −0.0742324
\(495\) −2.90470e11 −0.217459
\(496\) 7.76518e11 0.576081
\(497\) −3.59902e11 −0.264594
\(498\) −3.99305e11 −0.290919
\(499\) −2.49269e12 −1.79977 −0.899884 0.436130i \(-0.856349\pi\)
−0.899884 + 0.436130i \(0.856349\pi\)
\(500\) 1.77755e11 0.127191
\(501\) 1.13729e12 0.806498
\(502\) 1.32763e12 0.933063
\(503\) −7.65482e11 −0.533186 −0.266593 0.963809i \(-0.585898\pi\)
−0.266593 + 0.963809i \(0.585898\pi\)
\(504\) −1.30032e11 −0.0897661
\(505\) 3.91442e11 0.267828
\(506\) 3.28817e12 2.22986
\(507\) −8.73655e11 −0.587224
\(508\) 2.68930e11 0.179165
\(509\) 1.72021e12 1.13593 0.567966 0.823052i \(-0.307731\pi\)
0.567966 + 0.823052i \(0.307731\pi\)
\(510\) −2.78823e11 −0.182500
\(511\) −8.10203e10 −0.0525654
\(512\) 5.79868e11 0.372919
\(513\) −7.99094e11 −0.509413
\(514\) −9.24469e10 −0.0584196
\(515\) 2.84382e11 0.178143
\(516\) 2.31459e11 0.143731
\(517\) −1.57928e12 −0.972191
\(518\) −2.60396e11 −0.158910
\(519\) 1.16052e12 0.702100
\(520\) −4.36137e10 −0.0261582
\(521\) −2.13972e12 −1.27229 −0.636147 0.771568i \(-0.719473\pi\)
−0.636147 + 0.771568i \(0.719473\pi\)
\(522\) −2.08034e12 −1.22636
\(523\) 2.11674e12 1.23711 0.618557 0.785740i \(-0.287717\pi\)
0.618557 + 0.785740i \(0.287717\pi\)
\(524\) 1.55636e11 0.0901817
\(525\) −1.64264e11 −0.0943684
\(526\) 2.67786e12 1.52529
\(527\) 9.34430e11 0.527715
\(528\) 1.72997e12 0.968690
\(529\) 2.03581e12 1.13028
\(530\) −7.92054e11 −0.436027
\(531\) 1.63422e12 0.892044
\(532\) 4.26754e10 0.0230980
\(533\) 3.34507e11 0.179528
\(534\) −6.91540e11 −0.368029
\(535\) −3.94755e10 −0.0208322
\(536\) −2.37922e12 −1.24507
\(537\) 5.43873e11 0.282236
\(538\) −3.69962e12 −1.90387
\(539\) 2.58718e12 1.32031
\(540\) 1.27217e11 0.0643835
\(541\) 3.38496e12 1.69889 0.849445 0.527676i \(-0.176937\pi\)
0.849445 + 0.527676i \(0.176937\pi\)
\(542\) 2.23775e12 1.11382
\(543\) 7.99596e11 0.394704
\(544\) −1.15140e12 −0.563676
\(545\) −8.44282e11 −0.409924
\(546\) −2.98632e10 −0.0143804
\(547\) 2.73500e12 1.30622 0.653108 0.757265i \(-0.273465\pi\)
0.653108 + 0.757265i \(0.273465\pi\)
\(548\) 5.24350e11 0.248376
\(549\) −1.39351e12 −0.654687
\(550\) −3.07601e12 −1.43336
\(551\) −1.90358e12 −0.879810
\(552\) 1.57249e12 0.720877
\(553\) 5.10565e10 0.0232160
\(554\) −3.15907e12 −1.42484
\(555\) −2.78174e11 −0.124451
\(556\) 1.35456e11 0.0601122
\(557\) 4.19956e11 0.184865 0.0924326 0.995719i \(-0.470536\pi\)
0.0924326 + 0.995719i \(0.470536\pi\)
\(558\) −7.99477e11 −0.349102
\(559\) 2.67800e11 0.116000
\(560\) 1.16435e11 0.0500307
\(561\) 2.08177e12 0.887361
\(562\) 9.21626e11 0.389710
\(563\) −4.08232e12 −1.71245 −0.856227 0.516600i \(-0.827198\pi\)
−0.856227 + 0.516600i \(0.827198\pi\)
\(564\) 2.70883e11 0.112726
\(565\) −2.89608e11 −0.119562
\(566\) 1.16297e12 0.476315
\(567\) 2.41115e10 0.00979718
\(568\) −3.22300e12 −1.29925
\(569\) 3.56221e12 1.42467 0.712334 0.701840i \(-0.247638\pi\)
0.712334 + 0.701840i \(0.247638\pi\)
\(570\) 2.18285e11 0.0866139
\(571\) −1.37253e12 −0.540331 −0.270166 0.962814i \(-0.587078\pi\)
−0.270166 + 0.962814i \(0.587078\pi\)
\(572\) −1.16793e11 −0.0456179
\(573\) 1.54782e12 0.599825
\(574\) −6.95637e11 −0.267472
\(575\) −3.58939e12 −1.36935
\(576\) −1.04595e12 −0.395923
\(577\) −4.32295e12 −1.62364 −0.811818 0.583911i \(-0.801522\pi\)
−0.811818 + 0.583911i \(0.801522\pi\)
\(578\) −5.93984e11 −0.221360
\(579\) −2.41057e12 −0.891388
\(580\) 3.03054e11 0.111197
\(581\) 2.00657e11 0.0730570
\(582\) −8.29955e10 −0.0299848
\(583\) 5.91370e12 2.12007
\(584\) −7.25555e11 −0.258115
\(585\) 5.76448e10 0.0203497
\(586\) 2.12870e12 0.745719
\(587\) 5.59187e11 0.194395 0.0971977 0.995265i \(-0.469012\pi\)
0.0971977 + 0.995265i \(0.469012\pi\)
\(588\) −4.43760e11 −0.153091
\(589\) −7.31549e11 −0.250452
\(590\) −1.13989e12 −0.387282
\(591\) 1.26123e11 0.0425257
\(592\) −2.99360e12 −1.00172
\(593\) −1.93158e12 −0.641455 −0.320728 0.947171i \(-0.603927\pi\)
−0.320728 + 0.947171i \(0.603927\pi\)
\(594\) −4.54794e12 −1.49891
\(595\) 1.40113e11 0.0458302
\(596\) 3.93209e11 0.127648
\(597\) 1.69021e12 0.544575
\(598\) −6.52551e11 −0.208670
\(599\) 6.61008e11 0.209791 0.104895 0.994483i \(-0.466549\pi\)
0.104895 + 0.994483i \(0.466549\pi\)
\(600\) −1.47103e12 −0.463383
\(601\) −5.01901e12 −1.56922 −0.784608 0.619992i \(-0.787136\pi\)
−0.784608 + 0.619992i \(0.787136\pi\)
\(602\) −5.56914e11 −0.172824
\(603\) 3.14465e12 0.968599
\(604\) 1.16613e12 0.356518
\(605\) −6.93529e11 −0.210458
\(606\) 2.40026e12 0.722990
\(607\) 2.50351e12 0.748515 0.374257 0.927325i \(-0.377898\pi\)
0.374257 + 0.927325i \(0.377898\pi\)
\(608\) 9.01407e11 0.267519
\(609\) −5.78553e11 −0.170438
\(610\) 9.71985e11 0.284233
\(611\) 3.13414e11 0.0909774
\(612\) 6.45202e11 0.185915
\(613\) −2.29177e12 −0.655541 −0.327770 0.944757i \(-0.606297\pi\)
−0.327770 + 0.944757i \(0.606297\pi\)
\(614\) 3.92469e12 1.11442
\(615\) −7.43132e11 −0.209473
\(616\) −6.77180e11 −0.189492
\(617\) 1.35559e12 0.376568 0.188284 0.982115i \(-0.439707\pi\)
0.188284 + 0.982115i \(0.439707\pi\)
\(618\) 1.74378e12 0.480889
\(619\) −9.06881e11 −0.248280 −0.124140 0.992265i \(-0.539617\pi\)
−0.124140 + 0.992265i \(0.539617\pi\)
\(620\) 1.16464e11 0.0316540
\(621\) −5.30699e12 −1.43198
\(622\) 3.40686e12 0.912635
\(623\) 3.47510e11 0.0924211
\(624\) −3.43318e11 −0.0906497
\(625\) 3.12207e12 0.818431
\(626\) −5.98503e12 −1.55769
\(627\) −1.62978e12 −0.421139
\(628\) 1.39614e12 0.358187
\(629\) −3.60238e12 −0.917618
\(630\) −1.19877e11 −0.0303183
\(631\) −3.64544e12 −0.915415 −0.457708 0.889103i \(-0.651329\pi\)
−0.457708 + 0.889103i \(0.651329\pi\)
\(632\) 4.57223e11 0.113999
\(633\) 4.33855e12 1.07406
\(634\) 6.50019e12 1.59781
\(635\) −6.91252e11 −0.168715
\(636\) −1.01434e12 −0.245824
\(637\) −5.13435e11 −0.123554
\(638\) −1.08340e13 −2.58878
\(639\) 4.25989e12 1.01075
\(640\) 1.27318e12 0.299971
\(641\) −3.45800e12 −0.809029 −0.404515 0.914531i \(-0.632560\pi\)
−0.404515 + 0.914531i \(0.632560\pi\)
\(642\) −2.42058e11 −0.0562356
\(643\) 6.75236e12 1.55778 0.778890 0.627160i \(-0.215783\pi\)
0.778890 + 0.627160i \(0.215783\pi\)
\(644\) 2.83418e11 0.0649293
\(645\) −5.94937e11 −0.135348
\(646\) 2.82682e12 0.638633
\(647\) −2.74120e12 −0.614996 −0.307498 0.951549i \(-0.599492\pi\)
−0.307498 + 0.951549i \(0.599492\pi\)
\(648\) 2.15924e11 0.0481077
\(649\) 8.51071e12 1.88306
\(650\) 6.10447e11 0.134134
\(651\) −2.22339e11 −0.0485178
\(652\) −1.19995e12 −0.260046
\(653\) −5.45651e11 −0.117437 −0.0587186 0.998275i \(-0.518701\pi\)
−0.0587186 + 0.998275i \(0.518701\pi\)
\(654\) −5.17701e12 −1.10657
\(655\) −4.00043e11 −0.0849220
\(656\) −7.99729e12 −1.68607
\(657\) 9.58977e11 0.200800
\(658\) −6.51772e11 −0.135544
\(659\) 2.20632e12 0.455704 0.227852 0.973696i \(-0.426830\pi\)
0.227852 + 0.973696i \(0.426830\pi\)
\(660\) 2.59464e11 0.0532268
\(661\) 9.47272e12 1.93005 0.965024 0.262161i \(-0.0844353\pi\)
0.965024 + 0.262161i \(0.0844353\pi\)
\(662\) 1.46903e12 0.297282
\(663\) −4.13136e11 −0.0830389
\(664\) 1.79693e12 0.358736
\(665\) −1.09692e11 −0.0217509
\(666\) 3.08211e12 0.607036
\(667\) −1.26422e13 −2.47318
\(668\) 1.83565e12 0.356694
\(669\) −4.22117e11 −0.0814733
\(670\) −2.19342e12 −0.420519
\(671\) −7.25712e12 −1.38201
\(672\) 2.73964e11 0.0518241
\(673\) −3.22229e12 −0.605476 −0.302738 0.953074i \(-0.597901\pi\)
−0.302738 + 0.953074i \(0.597901\pi\)
\(674\) −4.78680e12 −0.893461
\(675\) 4.96457e12 0.920480
\(676\) −1.41012e12 −0.259715
\(677\) −2.90840e11 −0.0532115 −0.0266057 0.999646i \(-0.508470\pi\)
−0.0266057 + 0.999646i \(0.508470\pi\)
\(678\) −1.77583e12 −0.322751
\(679\) 4.17066e10 0.00752992
\(680\) 1.25474e12 0.225043
\(681\) −6.42613e11 −0.114495
\(682\) −4.16352e12 −0.736937
\(683\) −6.46343e12 −1.13650 −0.568250 0.822856i \(-0.692379\pi\)
−0.568250 + 0.822856i \(0.692379\pi\)
\(684\) −5.05117e11 −0.0882348
\(685\) −1.34778e12 −0.233890
\(686\) 2.16667e12 0.373538
\(687\) 3.22861e12 0.552980
\(688\) −6.40248e12 −1.08943
\(689\) −1.17360e12 −0.198396
\(690\) 1.44969e12 0.243475
\(691\) −1.94520e12 −0.324573 −0.162286 0.986744i \(-0.551887\pi\)
−0.162286 + 0.986744i \(0.551887\pi\)
\(692\) 1.87314e12 0.310522
\(693\) 8.95039e11 0.147415
\(694\) 2.23327e12 0.365447
\(695\) −3.48174e11 −0.0566063
\(696\) −5.18108e12 −0.836910
\(697\) −9.62362e12 −1.54451
\(698\) −1.21971e13 −1.94495
\(699\) 2.58596e12 0.409708
\(700\) −2.65131e11 −0.0417369
\(701\) −5.04036e12 −0.788370 −0.394185 0.919031i \(-0.628973\pi\)
−0.394185 + 0.919031i \(0.628973\pi\)
\(702\) 9.02557e11 0.140268
\(703\) 2.82024e12 0.435499
\(704\) −5.44711e12 −0.835775
\(705\) −6.96272e11 −0.106152
\(706\) 1.51376e12 0.229317
\(707\) −1.20617e12 −0.181561
\(708\) −1.45978e12 −0.218343
\(709\) −1.12096e13 −1.66602 −0.833012 0.553255i \(-0.813386\pi\)
−0.833012 + 0.553255i \(0.813386\pi\)
\(710\) −2.97131e12 −0.438819
\(711\) −6.04318e11 −0.0886854
\(712\) 3.11203e12 0.453821
\(713\) −4.85840e12 −0.704028
\(714\) 8.59151e11 0.123717
\(715\) 3.00203e11 0.0429574
\(716\) 8.77838e11 0.124826
\(717\) 5.38813e12 0.761381
\(718\) −1.68214e12 −0.236212
\(719\) −9.06205e12 −1.26458 −0.632290 0.774732i \(-0.717885\pi\)
−0.632290 + 0.774732i \(0.717885\pi\)
\(720\) −1.37815e12 −0.191118
\(721\) −8.76280e11 −0.120763
\(722\) 5.99590e12 0.821177
\(723\) −5.01063e12 −0.681977
\(724\) 1.29059e12 0.174568
\(725\) 1.18265e13 1.58977
\(726\) −4.25261e12 −0.568121
\(727\) 7.11638e12 0.944832 0.472416 0.881376i \(-0.343382\pi\)
0.472416 + 0.881376i \(0.343382\pi\)
\(728\) 1.34389e11 0.0177326
\(729\) 4.18017e12 0.548176
\(730\) −6.68895e11 −0.0871777
\(731\) −7.70449e12 −0.997966
\(732\) 1.24476e12 0.160246
\(733\) −5.13324e12 −0.656786 −0.328393 0.944541i \(-0.606507\pi\)
−0.328393 + 0.944541i \(0.606507\pi\)
\(734\) −1.09882e13 −1.39731
\(735\) 1.14063e12 0.144163
\(736\) 5.98647e12 0.752005
\(737\) 1.63767e13 2.04467
\(738\) 8.23374e12 1.02175
\(739\) 1.77858e12 0.219368 0.109684 0.993966i \(-0.465016\pi\)
0.109684 + 0.993966i \(0.465016\pi\)
\(740\) −4.48988e11 −0.0550417
\(741\) 3.23436e11 0.0394101
\(742\) 2.44060e12 0.295582
\(743\) 1.21010e13 1.45671 0.728354 0.685201i \(-0.240286\pi\)
0.728354 + 0.685201i \(0.240286\pi\)
\(744\) −1.99110e12 −0.238240
\(745\) −1.01070e12 −0.120204
\(746\) −5.96445e12 −0.705092
\(747\) −2.37503e12 −0.279078
\(748\) 3.36009e12 0.392458
\(749\) 1.21638e11 0.0141222
\(750\) −2.80163e12 −0.323321
\(751\) 1.05110e13 1.20577 0.602887 0.797827i \(-0.294017\pi\)
0.602887 + 0.797827i \(0.294017\pi\)
\(752\) −7.49300e12 −0.854428
\(753\) −4.37021e12 −0.495364
\(754\) 2.15005e12 0.242257
\(755\) −2.99740e12 −0.335725
\(756\) −3.92001e11 −0.0436455
\(757\) −7.38838e12 −0.817745 −0.408872 0.912592i \(-0.634078\pi\)
−0.408872 + 0.912592i \(0.634078\pi\)
\(758\) −4.35700e12 −0.479376
\(759\) −1.08238e13 −1.18383
\(760\) −9.82317e11 −0.106805
\(761\) 7.44732e12 0.804950 0.402475 0.915431i \(-0.368150\pi\)
0.402475 + 0.915431i \(0.368150\pi\)
\(762\) −4.23865e12 −0.455439
\(763\) 2.60153e12 0.277887
\(764\) 2.49826e12 0.265288
\(765\) −1.65841e12 −0.175072
\(766\) 1.47164e13 1.54444
\(767\) −1.68898e12 −0.176216
\(768\) 4.26767e12 0.442655
\(769\) −5.01390e12 −0.517019 −0.258510 0.966009i \(-0.583231\pi\)
−0.258510 + 0.966009i \(0.583231\pi\)
\(770\) −6.24298e11 −0.0640005
\(771\) 3.04310e11 0.0310150
\(772\) −3.89079e12 −0.394239
\(773\) −1.47690e13 −1.48779 −0.743896 0.668295i \(-0.767024\pi\)
−0.743896 + 0.668295i \(0.767024\pi\)
\(774\) 6.59178e12 0.660189
\(775\) 4.54492e12 0.452553
\(776\) 3.73492e11 0.0369746
\(777\) 8.57153e11 0.0843653
\(778\) −4.77458e12 −0.467226
\(779\) 7.53416e12 0.733021
\(780\) −5.14917e10 −0.00498094
\(781\) 2.21847e13 2.13365
\(782\) 1.87736e13 1.79522
\(783\) 1.74856e13 1.66247
\(784\) 1.22750e13 1.16038
\(785\) −3.58861e12 −0.337297
\(786\) −2.45300e12 −0.229243
\(787\) 2.41040e12 0.223977 0.111988 0.993710i \(-0.464278\pi\)
0.111988 + 0.993710i \(0.464278\pi\)
\(788\) 2.03569e11 0.0188081
\(789\) −8.81481e12 −0.809779
\(790\) 4.21517e11 0.0385029
\(791\) 8.92384e11 0.0810508
\(792\) 8.01528e12 0.723862
\(793\) 1.44020e12 0.129329
\(794\) 1.73195e12 0.154648
\(795\) 2.60723e12 0.231487
\(796\) 2.72809e12 0.240852
\(797\) 1.86970e13 1.64138 0.820690 0.571374i \(-0.193589\pi\)
0.820690 + 0.571374i \(0.193589\pi\)
\(798\) −6.72614e11 −0.0587155
\(799\) −9.01678e12 −0.782692
\(800\) −5.60021e12 −0.483392
\(801\) −4.11322e12 −0.353050
\(802\) 8.16773e12 0.697135
\(803\) 4.99416e12 0.423880
\(804\) −2.80898e12 −0.237081
\(805\) −7.28492e11 −0.0611425
\(806\) 8.26266e11 0.0689624
\(807\) 1.21782e13 1.01077
\(808\) −1.08015e13 −0.891528
\(809\) 5.77724e12 0.474189 0.237095 0.971487i \(-0.423805\pi\)
0.237095 + 0.971487i \(0.423805\pi\)
\(810\) 1.99062e11 0.0162483
\(811\) −2.29128e13 −1.85987 −0.929937 0.367719i \(-0.880139\pi\)
−0.929937 + 0.367719i \(0.880139\pi\)
\(812\) −9.33815e11 −0.0753805
\(813\) −7.36609e12 −0.591329
\(814\) 1.60510e13 1.28143
\(815\) 3.08434e12 0.244879
\(816\) 9.87711e12 0.779874
\(817\) 6.03171e12 0.473632
\(818\) −2.63182e13 −2.05526
\(819\) −1.77624e11 −0.0137951
\(820\) −1.19945e12 −0.0926447
\(821\) −1.20930e13 −0.928944 −0.464472 0.885588i \(-0.653756\pi\)
−0.464472 + 0.885588i \(0.653756\pi\)
\(822\) −8.26437e12 −0.631374
\(823\) 1.45424e13 1.10493 0.552467 0.833535i \(-0.313687\pi\)
0.552467 + 0.833535i \(0.313687\pi\)
\(824\) −7.84729e12 −0.592990
\(825\) 1.01254e13 0.760974
\(826\) 3.51239e12 0.262538
\(827\) −1.27869e13 −0.950584 −0.475292 0.879828i \(-0.657658\pi\)
−0.475292 + 0.879828i \(0.657658\pi\)
\(828\) −3.35461e12 −0.248031
\(829\) 2.02455e13 1.48879 0.744394 0.667741i \(-0.232739\pi\)
0.744394 + 0.667741i \(0.232739\pi\)
\(830\) 1.65660e12 0.121162
\(831\) 1.03988e13 0.756448
\(832\) 1.08100e12 0.0782116
\(833\) 1.47713e13 1.06296
\(834\) −2.13495e12 −0.152806
\(835\) −4.71832e12 −0.335891
\(836\) −2.63055e12 −0.186260
\(837\) 6.71975e12 0.473248
\(838\) 2.10280e13 1.47299
\(839\) −1.34126e13 −0.934509 −0.467254 0.884123i \(-0.654757\pi\)
−0.467254 + 0.884123i \(0.654757\pi\)
\(840\) −2.98555e11 −0.0206903
\(841\) 2.71467e13 1.87126
\(842\) −1.79150e13 −1.22832
\(843\) −3.03374e12 −0.206897
\(844\) 7.00265e12 0.475030
\(845\) 3.62455e12 0.244568
\(846\) 7.71455e12 0.517778
\(847\) 2.13701e12 0.142669
\(848\) 2.80580e13 1.86327
\(849\) −3.82818e12 −0.252876
\(850\) −1.75623e13 −1.15397
\(851\) 1.87299e13 1.22420
\(852\) −3.80518e12 −0.247399
\(853\) 2.07726e13 1.34344 0.671722 0.740803i \(-0.265555\pi\)
0.671722 + 0.740803i \(0.265555\pi\)
\(854\) −2.99503e12 −0.192682
\(855\) 1.29834e12 0.0830887
\(856\) 1.08930e12 0.0693448
\(857\) 1.47520e13 0.934192 0.467096 0.884207i \(-0.345300\pi\)
0.467096 + 0.884207i \(0.345300\pi\)
\(858\) 1.84080e12 0.115961
\(859\) 1.11817e13 0.700711 0.350356 0.936617i \(-0.386061\pi\)
0.350356 + 0.936617i \(0.386061\pi\)
\(860\) −9.60260e11 −0.0598612
\(861\) 2.28985e12 0.142001
\(862\) −7.00725e12 −0.432280
\(863\) −1.23584e13 −0.758428 −0.379214 0.925309i \(-0.623806\pi\)
−0.379214 + 0.925309i \(0.623806\pi\)
\(864\) −8.28002e12 −0.505498
\(865\) −4.81467e12 −0.292412
\(866\) −2.56692e10 −0.00155089
\(867\) 1.95523e12 0.117520
\(868\) −3.58866e11 −0.0214582
\(869\) −3.14717e12 −0.187211
\(870\) −4.77648e12 −0.282665
\(871\) −3.25002e12 −0.191339
\(872\) 2.32973e13 1.36453
\(873\) −4.93650e11 −0.0287644
\(874\) −1.46975e13 −0.852005
\(875\) 1.40786e12 0.0811940
\(876\) −8.56614e11 −0.0491492
\(877\) 1.39608e13 0.796917 0.398459 0.917186i \(-0.369545\pi\)
0.398459 + 0.917186i \(0.369545\pi\)
\(878\) −2.09676e13 −1.19076
\(879\) −7.00711e12 −0.395903
\(880\) −7.17715e12 −0.403441
\(881\) −1.27684e13 −0.714079 −0.357039 0.934089i \(-0.616214\pi\)
−0.357039 + 0.934089i \(0.616214\pi\)
\(882\) −1.26380e13 −0.703183
\(883\) 8.68951e12 0.481030 0.240515 0.970645i \(-0.422684\pi\)
0.240515 + 0.970645i \(0.422684\pi\)
\(884\) −6.66822e11 −0.0367261
\(885\) 3.75220e12 0.205608
\(886\) 3.67894e13 2.00572
\(887\) 8.63030e11 0.0468133 0.0234067 0.999726i \(-0.492549\pi\)
0.0234067 + 0.999726i \(0.492549\pi\)
\(888\) 7.67601e12 0.414264
\(889\) 2.12999e12 0.114372
\(890\) 2.86901e12 0.153277
\(891\) −1.48626e12 −0.0790031
\(892\) −6.81319e11 −0.0360337
\(893\) 7.05908e12 0.371464
\(894\) −6.19744e12 −0.324484
\(895\) −2.25638e12 −0.117546
\(896\) −3.92311e12 −0.203350
\(897\) 2.14802e12 0.110783
\(898\) 5.51175e12 0.282844
\(899\) 1.60076e13 0.817350
\(900\) 3.13816e12 0.159435
\(901\) 3.37639e13 1.70683
\(902\) 4.28797e13 2.15686
\(903\) 1.83321e12 0.0917524
\(904\) 7.99151e12 0.397989
\(905\) −3.31730e12 −0.164387
\(906\) −1.83796e13 −0.906272
\(907\) −7.79700e12 −0.382556 −0.191278 0.981536i \(-0.561263\pi\)
−0.191278 + 0.981536i \(0.561263\pi\)
\(908\) −1.03721e12 −0.0506385
\(909\) 1.42766e13 0.693563
\(910\) 1.23894e11 0.00598915
\(911\) −3.57204e13 −1.71824 −0.859120 0.511774i \(-0.828988\pi\)
−0.859120 + 0.511774i \(0.828988\pi\)
\(912\) −7.73261e12 −0.370126
\(913\) −1.23687e13 −0.589122
\(914\) 1.83424e13 0.869356
\(915\) −3.19951e12 −0.150900
\(916\) 5.21114e12 0.244570
\(917\) 1.23267e12 0.0575686
\(918\) −2.59662e13 −1.20675
\(919\) −1.55458e12 −0.0718939 −0.0359470 0.999354i \(-0.511445\pi\)
−0.0359470 + 0.999354i \(0.511445\pi\)
\(920\) −6.52381e12 −0.300232
\(921\) −1.29190e13 −0.591645
\(922\) 4.97315e12 0.226643
\(923\) −4.40264e12 −0.199666
\(924\) −7.99501e11 −0.0360824
\(925\) −1.75214e13 −0.786922
\(926\) −3.77265e13 −1.68615
\(927\) 1.03719e13 0.461316
\(928\) −1.97244e13 −0.873049
\(929\) 2.73767e13 1.20590 0.602948 0.797781i \(-0.293993\pi\)
0.602948 + 0.797781i \(0.293993\pi\)
\(930\) −1.83561e12 −0.0804649
\(931\) −1.15642e13 −0.504477
\(932\) 4.17387e12 0.181204
\(933\) −1.12145e13 −0.484519
\(934\) 2.93006e13 1.25984
\(935\) −8.63670e12 −0.369569
\(936\) −1.59066e12 −0.0677388
\(937\) −3.01208e13 −1.27655 −0.638276 0.769808i \(-0.720352\pi\)
−0.638276 + 0.769808i \(0.720352\pi\)
\(938\) 6.75870e12 0.285069
\(939\) 1.97011e13 0.826981
\(940\) −1.12382e12 −0.0469484
\(941\) −3.77327e13 −1.56879 −0.784395 0.620261i \(-0.787027\pi\)
−0.784395 + 0.620261i \(0.787027\pi\)
\(942\) −2.20048e13 −0.910517
\(943\) 5.00362e13 2.06054
\(944\) 4.03797e13 1.65497
\(945\) 1.00759e12 0.0411000
\(946\) 3.43287e13 1.39363
\(947\) −2.42971e13 −0.981703 −0.490852 0.871243i \(-0.663314\pi\)
−0.490852 + 0.871243i \(0.663314\pi\)
\(948\) 5.39812e11 0.0217073
\(949\) −9.91111e11 −0.0396666
\(950\) 1.37492e13 0.547672
\(951\) −2.13969e13 −0.848278
\(952\) −3.86631e12 −0.152556
\(953\) 3.11007e13 1.22138 0.610691 0.791869i \(-0.290892\pi\)
0.610691 + 0.791869i \(0.290892\pi\)
\(954\) −2.88876e13 −1.12913
\(955\) −6.42148e12 −0.249816
\(956\) 8.69672e12 0.336740
\(957\) 3.56626e13 1.37439
\(958\) −3.21926e13 −1.23484
\(959\) 4.15298e12 0.158554
\(960\) −2.40152e12 −0.0912569
\(961\) −2.02879e13 −0.767328
\(962\) −3.18539e12 −0.119915
\(963\) −1.43974e12 −0.0539468
\(964\) −8.08742e12 −0.301622
\(965\) 1.00008e13 0.371246
\(966\) −4.46700e12 −0.165051
\(967\) 1.88582e13 0.693554 0.346777 0.937948i \(-0.387276\pi\)
0.346777 + 0.937948i \(0.387276\pi\)
\(968\) 1.91374e13 0.700557
\(969\) −9.30512e12 −0.339051
\(970\) 3.44325e11 0.0124881
\(971\) −2.84505e13 −1.02708 −0.513539 0.858066i \(-0.671666\pi\)
−0.513539 + 0.858066i \(0.671666\pi\)
\(972\) 7.46257e12 0.268158
\(973\) 1.07285e12 0.0383734
\(974\) −2.79219e12 −0.0994099
\(975\) −2.00943e12 −0.0712117
\(976\) −3.44319e13 −1.21461
\(977\) −1.03143e13 −0.362172 −0.181086 0.983467i \(-0.557961\pi\)
−0.181086 + 0.983467i \(0.557961\pi\)
\(978\) 1.89127e13 0.661040
\(979\) −2.14208e13 −0.745271
\(980\) 1.84104e12 0.0637596
\(981\) −3.07924e13 −1.06153
\(982\) −5.59559e13 −1.92019
\(983\) −9.89411e12 −0.337976 −0.168988 0.985618i \(-0.554050\pi\)
−0.168988 + 0.985618i \(0.554050\pi\)
\(984\) 2.05061e13 0.697278
\(985\) −5.23251e11 −0.0177111
\(986\) −6.18558e13 −2.08418
\(987\) 2.14546e12 0.0719603
\(988\) 5.22043e11 0.0174301
\(989\) 4.00581e13 1.33139
\(990\) 7.38935e12 0.244483
\(991\) −4.83417e13 −1.59217 −0.796087 0.605182i \(-0.793100\pi\)
−0.796087 + 0.605182i \(0.793100\pi\)
\(992\) −7.58012e12 −0.248527
\(993\) −4.83565e12 −0.157828
\(994\) 9.15566e12 0.297475
\(995\) −7.01224e12 −0.226805
\(996\) 2.12151e12 0.0683092
\(997\) 2.17418e13 0.696894 0.348447 0.937329i \(-0.386709\pi\)
0.348447 + 0.937329i \(0.386709\pi\)
\(998\) 6.34125e13 2.02343
\(999\) −2.59058e13 −0.822909
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 197.10.a.b.1.20 76
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
197.10.a.b.1.20 76 1.1 even 1 trivial